TSTP Solution File: NUM501+3 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : NUM501+3 : TPTP v8.1.2. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n017.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Thu Aug 31 11:56:39 EDT 2023
% Result : Theorem 278.86s 35.84s
% Output : Proof 278.86s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.06/0.12 % Problem : NUM501+3 : TPTP v8.1.2. Released v4.0.0.
% 0.06/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.33 % Computer : n017.cluster.edu
% 0.13/0.33 % Model : x86_64 x86_64
% 0.13/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.33 % Memory : 8042.1875MB
% 0.13/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Fri Aug 25 17:16:25 EDT 2023
% 0.13/0.34 % CPUTime :
% 278.86/35.84 Command-line arguments: --no-flatten-goal
% 278.86/35.84
% 278.86/35.84 % SZS status Theorem
% 278.86/35.84
% 278.86/35.85 % SZS output start Proof
% 278.86/35.85 Take the following subset of the input axioms:
% 278.86/35.85 fof(mDefPrime, definition, ![W0]: (aNaturalNumber0(W0) => (isPrime0(W0) <=> (W0!=sz00 & (W0!=sz10 & ![W1]: ((aNaturalNumber0(W1) & doDivides0(W1, W0)) => (W1=sz10 | W1=W0))))))).
% 278.86/35.85 fof(mMulAsso, axiom, ![W2, W1_2, W0_2]: ((aNaturalNumber0(W0_2) & (aNaturalNumber0(W1_2) & aNaturalNumber0(W2))) => sdtasdt0(sdtasdt0(W0_2, W1_2), W2)=sdtasdt0(W0_2, sdtasdt0(W1_2, W2)))).
% 278.86/35.85 fof(mMulComm, axiom, ![W1_2, W0_2]: ((aNaturalNumber0(W0_2) & aNaturalNumber0(W1_2)) => sdtasdt0(W0_2, W1_2)=sdtasdt0(W1_2, W0_2))).
% 278.86/35.85 fof(mSortsB_02, axiom, ![W1_2, W0_2]: ((aNaturalNumber0(W0_2) & aNaturalNumber0(W1_2)) => aNaturalNumber0(sdtasdt0(W0_2, W1_2)))).
% 278.86/35.85 fof(m__, conjecture, ?[W0_2]: (aNaturalNumber0(W0_2) & sdtasdt0(xn, xm)=sdtasdt0(xr, W0_2)) | doDivides0(xr, sdtasdt0(xn, xm))).
% 278.86/35.85 fof(m__1799, hypothesis, ![W1_2, W0_2, W2_2]: ((aNaturalNumber0(W0_2) & (aNaturalNumber0(W1_2) & aNaturalNumber0(W2_2))) => ((((W2_2!=sz00 & (W2_2!=sz10 & ![W3]: ((aNaturalNumber0(W3) & (?[W4]: (aNaturalNumber0(W4) & W2_2=sdtasdt0(W3, W4)) & doDivides0(W3, W2_2))) => (W3=sz10 | W3=W2_2)))) | isPrime0(W2_2)) & (?[W3_2]: (aNaturalNumber0(W3_2) & sdtasdt0(W0_2, W1_2)=sdtasdt0(W2_2, W3_2)) | doDivides0(W2_2, sdtasdt0(W0_2, W1_2)))) => (iLess0(sdtpldt0(sdtpldt0(W0_2, W1_2), W2_2), sdtpldt0(sdtpldt0(xn, xm), xp)) => ((?[W3_2]: (aNaturalNumber0(W3_2) & W0_2=sdtasdt0(W2_2, W3_2)) & doDivides0(W2_2, W0_2)) | (?[W3_2]: (aNaturalNumber0(W3_2) & W1_2=sdtasdt0(W2_2, W3_2)) & doDivides0(W2_2, W1_2))))))).
% 278.86/35.85 fof(m__1837, hypothesis, aNaturalNumber0(xn) & (aNaturalNumber0(xm) & aNaturalNumber0(xp))).
% 278.86/35.85 fof(m__1860, hypothesis, xp!=sz00 & (xp!=sz10 & (![W0_2]: ((aNaturalNumber0(W0_2) & (?[W1_2]: (aNaturalNumber0(W1_2) & xp=sdtasdt0(W0_2, W1_2)) | doDivides0(W0_2, xp))) => (W0_2=sz10 | W0_2=xp)) & (isPrime0(xp) & (?[W0_2]: (aNaturalNumber0(W0_2) & sdtasdt0(xn, xm)=sdtasdt0(xp, W0_2)) & doDivides0(xp, sdtasdt0(xn, xm))))))).
% 278.86/35.85 fof(m__1870, hypothesis, ~(?[W0_2]: (aNaturalNumber0(W0_2) & sdtpldt0(xp, W0_2)=xn) | sdtlseqdt0(xp, xn))).
% 278.86/35.85 fof(m__2075, hypothesis, ~(?[W0_2]: (aNaturalNumber0(W0_2) & sdtpldt0(xp, W0_2)=xm) | sdtlseqdt0(xp, xm))).
% 278.86/35.85 fof(m__2306, hypothesis, aNaturalNumber0(xk) & (sdtasdt0(xn, xm)=sdtasdt0(xp, xk) & xk=sdtsldt0(sdtasdt0(xn, xm), xp))).
% 278.86/35.85 fof(m__2342, hypothesis, aNaturalNumber0(xr) & (?[W0_2]: (aNaturalNumber0(W0_2) & xk=sdtasdt0(xr, W0_2)) & (doDivides0(xr, xk) & (xr!=sz00 & (xr!=sz10 & (![W0_2]: ((aNaturalNumber0(W0_2) & (?[W1_2]: (aNaturalNumber0(W1_2) & xr=sdtasdt0(W0_2, W1_2)) | doDivides0(W0_2, xr))) => (W0_2=sz10 | W0_2=xr)) & isPrime0(xr))))))).
% 278.86/35.85
% 278.86/35.85 Now clausify the problem and encode Horn clauses using encoding 3 of
% 278.86/35.85 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 278.86/35.85 We repeatedly replace C & s=t => u=v by the two clauses:
% 278.86/35.85 fresh(y, y, x1...xn) = u
% 278.86/35.85 C => fresh(s, t, x1...xn) = v
% 278.86/35.85 where fresh is a fresh function symbol and x1..xn are the free
% 278.86/35.85 variables of u and v.
% 278.86/35.85 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 278.86/35.85 input problem has no model of domain size 1).
% 278.86/35.85
% 278.86/35.85 The encoding turns the above axioms into the following unit equations and goals:
% 278.86/35.85
% 278.86/35.85 Axiom 1 (m__1860_1): aNaturalNumber0(w0_4) = true2.
% 278.86/35.85 Axiom 2 (m__2342_2): aNaturalNumber0(w0) = true2.
% 278.86/35.85 Axiom 3 (m__2342_1): aNaturalNumber0(xr) = true2.
% 278.86/35.85 Axiom 4 (m__1837_2): aNaturalNumber0(xp) = true2.
% 278.86/35.85 Axiom 5 (m__2342): xk = sdtasdt0(xr, w0).
% 278.86/35.85 Axiom 6 (m__1860): sdtasdt0(xn, xm) = sdtasdt0(xp, w0_4).
% 278.86/35.85 Axiom 7 (m__2306): sdtasdt0(xn, xm) = sdtasdt0(xp, xk).
% 278.86/35.85 Axiom 8 (mMulComm): fresh20(X, X, Y, Z) = sdtasdt0(Y, Z).
% 278.86/35.85 Axiom 9 (mMulComm): fresh19(X, X, Y, Z) = sdtasdt0(Z, Y).
% 278.86/35.85 Axiom 10 (mSortsB_02): fresh16(X, X, Y, Z) = aNaturalNumber0(sdtasdt0(Y, Z)).
% 278.86/35.85 Axiom 11 (mSortsB_02): fresh15(X, X, Y, Z) = true2.
% 278.86/35.85 Axiom 12 (mMulAsso): fresh107(X, X, Y, Z, W) = sdtasdt0(Y, sdtasdt0(Z, W)).
% 278.86/35.86 Axiom 13 (mMulAsso): fresh21(X, X, Y, Z, W) = sdtasdt0(sdtasdt0(Y, Z), W).
% 278.86/35.86 Axiom 14 (mMulComm): fresh20(aNaturalNumber0(X), true2, Y, X) = fresh19(aNaturalNumber0(Y), true2, Y, X).
% 278.86/35.86 Axiom 15 (mSortsB_02): fresh16(aNaturalNumber0(X), true2, Y, X) = fresh15(aNaturalNumber0(Y), true2, Y, X).
% 278.86/35.86 Axiom 16 (mMulAsso): fresh106(X, X, Y, Z, W) = fresh107(aNaturalNumber0(Y), true2, Y, Z, W).
% 278.86/35.86 Axiom 17 (mMulAsso): fresh106(aNaturalNumber0(X), true2, Y, Z, X) = fresh21(aNaturalNumber0(Z), true2, Y, Z, X).
% 278.86/35.86
% 278.86/35.86 Lemma 18: sdtasdt0(xn, xm) = sdtasdt0(w0_4, xp).
% 278.86/35.86 Proof:
% 278.86/35.86 sdtasdt0(xn, xm)
% 278.86/35.86 = { by axiom 6 (m__1860) }
% 278.86/35.86 sdtasdt0(xp, w0_4)
% 278.86/35.86 = { by axiom 9 (mMulComm) R->L }
% 278.86/35.86 fresh19(true2, true2, w0_4, xp)
% 278.86/35.86 = { by axiom 1 (m__1860_1) R->L }
% 278.86/35.86 fresh19(aNaturalNumber0(w0_4), true2, w0_4, xp)
% 278.86/35.86 = { by axiom 14 (mMulComm) R->L }
% 278.86/35.86 fresh20(aNaturalNumber0(xp), true2, w0_4, xp)
% 278.86/35.86 = { by axiom 4 (m__1837_2) }
% 278.86/35.86 fresh20(true2, true2, w0_4, xp)
% 278.86/35.86 = { by axiom 8 (mMulComm) }
% 278.86/35.86 sdtasdt0(w0_4, xp)
% 278.86/35.86
% 278.86/35.86 Lemma 19: aNaturalNumber0(sdtasdt0(w0, xp)) = true2.
% 278.86/35.86 Proof:
% 278.86/35.86 aNaturalNumber0(sdtasdt0(w0, xp))
% 278.86/35.86 = { by axiom 10 (mSortsB_02) R->L }
% 278.86/35.86 fresh16(true2, true2, w0, xp)
% 278.86/35.86 = { by axiom 4 (m__1837_2) R->L }
% 278.86/35.86 fresh16(aNaturalNumber0(xp), true2, w0, xp)
% 278.86/35.86 = { by axiom 15 (mSortsB_02) }
% 278.86/35.86 fresh15(aNaturalNumber0(w0), true2, w0, xp)
% 278.86/35.86 = { by axiom 2 (m__2342_2) }
% 278.86/35.86 fresh15(true2, true2, w0, xp)
% 278.86/35.86 = { by axiom 11 (mSortsB_02) }
% 278.86/35.86 true2
% 278.86/35.86
% 278.86/35.86 Lemma 20: fresh20(aNaturalNumber0(X), true2, w0, X) = sdtasdt0(X, w0).
% 278.86/35.86 Proof:
% 278.86/35.86 fresh20(aNaturalNumber0(X), true2, w0, X)
% 278.86/35.86 = { by axiom 14 (mMulComm) }
% 278.86/35.86 fresh19(aNaturalNumber0(w0), true2, w0, X)
% 278.86/35.86 = { by axiom 2 (m__2342_2) }
% 278.86/35.86 fresh19(true2, true2, w0, X)
% 278.86/35.86 = { by axiom 9 (mMulComm) }
% 278.86/35.86 sdtasdt0(X, w0)
% 278.86/35.86
% 278.86/35.86 Goal 1 (m__): tuple3(sdtasdt0(xn, xm), aNaturalNumber0(X)) = tuple3(sdtasdt0(xr, X), true2).
% 278.86/35.86 The goal is true when:
% 278.86/35.86 X = sdtasdt0(w0, xp)
% 278.86/35.86
% 278.86/35.86 Proof:
% 278.86/35.86 tuple3(sdtasdt0(xn, xm), aNaturalNumber0(sdtasdt0(w0, xp)))
% 278.86/35.86 = { by lemma 18 }
% 278.86/35.86 tuple3(sdtasdt0(w0_4, xp), aNaturalNumber0(sdtasdt0(w0, xp)))
% 278.86/35.86 = { by lemma 19 }
% 278.86/35.86 tuple3(sdtasdt0(w0_4, xp), true2)
% 278.86/35.86 = { by lemma 18 R->L }
% 278.86/35.86 tuple3(sdtasdt0(xn, xm), true2)
% 278.86/35.86 = { by axiom 7 (m__2306) }
% 278.86/35.86 tuple3(sdtasdt0(xp, xk), true2)
% 278.86/35.86 = { by axiom 5 (m__2342) }
% 278.86/35.86 tuple3(sdtasdt0(xp, sdtasdt0(xr, w0)), true2)
% 278.86/35.86 = { by lemma 20 R->L }
% 278.86/35.86 tuple3(sdtasdt0(xp, fresh20(aNaturalNumber0(xr), true2, w0, xr)), true2)
% 278.86/35.86 = { by axiom 3 (m__2342_1) }
% 278.86/35.86 tuple3(sdtasdt0(xp, fresh20(true2, true2, w0, xr)), true2)
% 278.86/35.86 = { by axiom 8 (mMulComm) }
% 278.86/35.86 tuple3(sdtasdt0(xp, sdtasdt0(w0, xr)), true2)
% 278.86/35.86 = { by axiom 12 (mMulAsso) R->L }
% 278.86/35.86 tuple3(fresh107(true2, true2, xp, w0, xr), true2)
% 278.86/35.86 = { by axiom 4 (m__1837_2) R->L }
% 278.86/35.86 tuple3(fresh107(aNaturalNumber0(xp), true2, xp, w0, xr), true2)
% 278.86/35.86 = { by axiom 16 (mMulAsso) R->L }
% 278.86/35.86 tuple3(fresh106(true2, true2, xp, w0, xr), true2)
% 278.86/35.86 = { by axiom 3 (m__2342_1) R->L }
% 278.86/35.86 tuple3(fresh106(aNaturalNumber0(xr), true2, xp, w0, xr), true2)
% 278.86/35.86 = { by axiom 17 (mMulAsso) }
% 278.86/35.86 tuple3(fresh21(aNaturalNumber0(w0), true2, xp, w0, xr), true2)
% 278.86/35.86 = { by axiom 2 (m__2342_2) }
% 278.86/35.86 tuple3(fresh21(true2, true2, xp, w0, xr), true2)
% 278.86/35.86 = { by axiom 13 (mMulAsso) }
% 278.86/35.86 tuple3(sdtasdt0(sdtasdt0(xp, w0), xr), true2)
% 278.86/35.86 = { by lemma 20 R->L }
% 278.86/35.86 tuple3(sdtasdt0(fresh20(aNaturalNumber0(xp), true2, w0, xp), xr), true2)
% 278.86/35.86 = { by axiom 4 (m__1837_2) }
% 278.86/35.86 tuple3(sdtasdt0(fresh20(true2, true2, w0, xp), xr), true2)
% 278.86/35.86 = { by axiom 8 (mMulComm) }
% 278.86/35.86 tuple3(sdtasdt0(sdtasdt0(w0, xp), xr), true2)
% 278.86/35.86 = { by axiom 9 (mMulComm) R->L }
% 278.86/35.86 tuple3(fresh19(true2, true2, xr, sdtasdt0(w0, xp)), true2)
% 278.86/35.86 = { by axiom 3 (m__2342_1) R->L }
% 278.86/35.86 tuple3(fresh19(aNaturalNumber0(xr), true2, xr, sdtasdt0(w0, xp)), true2)
% 278.86/35.86 = { by axiom 14 (mMulComm) R->L }
% 278.86/35.86 tuple3(fresh20(aNaturalNumber0(sdtasdt0(w0, xp)), true2, xr, sdtasdt0(w0, xp)), true2)
% 278.86/35.86 = { by lemma 19 }
% 278.86/35.86 tuple3(fresh20(true2, true2, xr, sdtasdt0(w0, xp)), true2)
% 278.86/35.86 = { by axiom 8 (mMulComm) }
% 278.86/35.86 tuple3(sdtasdt0(xr, sdtasdt0(w0, xp)), true2)
% 278.86/35.86 % SZS output end Proof
% 278.86/35.86
% 278.86/35.86 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------